In this slightly longer post, I wanted to explore some of the more subtle ideas of representation, and what it means to represent a quantum state vector in a coordinate system (basis) like position.
Position is easiest to think about conceptually, but the momentum and energy eigenstates are also valid basis states that you can use to represent quantum states. Of course qubits, which are simple two dimensional quantum states, are represented in computational basis states, and when we begin exploring qubits more closely the maths will be simpler.
For this week however, I wanted to explore the ideas of what makes a valid, measurable quantum state, so we to look at ideas of orthonormality and completeness.
Are there any concepts you would like to see expanded on or explored in quantum mechanics?
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#physics #quantummechanics #fyp #science #quantum
2 months agojclar121
One detail, it is technically not true to say that the position eigenstates form basis vectors in the Hilbert space. This is because the position operator is unbounded and discontinuous in the operator topology induced by the bra-ket inner product. Thus, it has no eigenvectors at all! However, if we extend the Hilbert space of wavefunctions to include distributions, then the position operator will have eigenvectors in this "rigged" Hibert space, which in the position representation are Dirac-delta distributions. I think this is important to point out because this state of affairs is typically assumed in quantum mechanics courses, but it is often a detail not discussed with enough attention. Even though it comes up quite often in quantum field theory when you want to discuss, e.g., field operators.
3 months agoarikpratama14
That's beautiful design and visualization for a complex topic like Quantum Mechanics 😮👏. May I know what tools do you use to create the slides?
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